Understanding Splines: Solving with Tridiagonal Matrices

[600burger]

I'm trying to find a spline for a set of points and solve it with a tridiagonal (at least that's what my prof wants). But...

1.)I'm unfamiliar with how splines work
2.)I don't see how a tridiagonal matrix will fit in
3.)what will the "Solution" look like? What will the solution of the tridiagonal kick out?

I have read the wolfram site and wikipedia as well as many other sites on the issue.

I really just need a general explanation of how splines are solved. I.E. not exact equations to find coefficients.

Thanks all!

 

[fresh_42]

For the piecewise polynomial function we have
$$
p(u)=\sum_{j=0}^{n-2}\sum_{i=0}^d c_{ij}(u-\tau_j)_+^i
$$
with the intermediate points $\tau_j$ and the function $(u)_+^i = \begin{cases} 0 & \text{ if } u<0 \\ u^i &\text{ if }u \geq 0 \end{cases}$
I assume that the $(c_{ij})$ are supposed to be triagonal.